The Flaw in Ampere's Law and the Displacement Current
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You should be comfortable with the divergence and curl operators in vector calculus, and you must understand the continuity equation for charge density and current density.
Why this matters
Maxwell’s displacement current resolves a fundamental contradiction between Ampère’s law and charge conservation.
In the previous lecture, we treated magnetic fields as generated solely by steady currents. However, this static assumption breaks down the moment charges begin to accumulate or de-accumulate in space. If we strictly apply the original Ampère law to a time-varying system, we arrive at a mathematical impossibility that violates the conservation of charge. This lecture identifies that flaw and introduces the displacement current, a term that not only fixes the inconsistency but also predicts the existence of electromagnetic waves, fundamentally changing our understanding of light.
The Logical Conflict
Imagine a simple circuit with a capacitor being charged. Current flows through the wires, but there is no physical current crossing the gap between the capacitor plates. If we draw a surface that passes through the wire, Ampère’s law tells us the circulation of the magnetic field depends on that current. But if we draw a different surface that passes through the gap, the current is zero, suggesting the magnetic circulation should be zero. This is a paradox: the magnetic field should not depend on which imaginary surface we choose. The resolution is that the changing electric field in the gap acts as a source of the magnetic field, just as the current in the wire does. This "effective current" is the displacement current.
Mathematical Inconsistency in Static Ampère Law
The magnetostatic form of Ampère’s law is given by:
To see where this fails, we take the divergence of both sides. A fundamental vector identity states that the divergence of a curl is always zero:
Therefore, the right-hand side must also have zero divergence:
However, the continuity equation, which expresses the conservation of charge, states:
These two equations are only compatible if \(\frac{\partial \rho}{\partial t} = 0\), meaning the charge density is constant in time. In any dynamic system, such as a charging capacitor, \(\rho\) changes with time. Thus, the original Ampère’s law is mathematically inconsistent with the conservation of charge whenever charge densities are time-dependent.
Deriving the Displacement Current Term
To fix this, we require that the divergence of the right-hand side of Ampère’s law matches the continuity equation. We know from Gauss’s law that \(\nabla \cdot \mathbf{E} = \rho / \varepsilon_0\). Taking the time derivative of Gauss’s law gives:
Rearranging the continuity equation, we have \(\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t}\). Substituting this into the expression above:
This suggests that if we add a term \(\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\) to the current density, the divergence of the total current will be zero. Let us define the total current density as \(\mathbf{J}_{\text{total}} = \mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\). Then:
Thus, the corrected Ampère-Maxwell law is:
The term \(\mathbf{J}_d = \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\) is called the displacement current density. It has the same units as current density and acts as a source for the magnetic field.
Physical Interpretation and Symmetry
The displacement current is not a flow of charges; it is a term arising from the time-varying electric field. In the case of a charging capacitor, the electric field between the plates changes as charge accumulates. This changing \(\mathbf{E}\)-field generates a magnetic field in the gap, exactly as if a current were flowing. This restores the symmetry between electric and magnetic fields: just as a changing magnetic field creates an electric field (Faraday’s law), a changing electric field creates a magnetic field (Ampère-Maxwell law). This symmetry is crucial for the propagation of electromagnetic waves, as it allows energy to be transferred between the electric and magnetic fields as they propagate through space.
The derivation
The Displacement Current from Charge Conservation
The magnetostatic Ampère law ∇×B = μ0J forces ∇·J = 0, which contradicts the continuity equation ∇·J + ∂ρ/∂t = 0 whenever the charge density is time-dependent. Requiring consistency with continuity and with Gauss's law ∇·E = ρ/ε0 forces the addition of exactly one extra term, Maxwell's displacement current ε0 ∂E/∂t, giving the corrected law ∇×B = μ0(J + ε0 ∂E/∂t).
A common misconception is that displacement current involves the actual movement of charges. It does not. It is a mathematical term that accounts for the time-varying electric field. In vacuum, where \(\mathbf{J} = 0\), the displacement current is the sole source of the magnetic field. This is why electromagnetic waves can propagate through empty space, even though no charges are moving. C
Worked example
Problem. A parallel-plate capacitor with plate area \(A = 0.01\ \mathrm{m^2}\) is being charged by a current \(I = 5\ \mathrm{A}\). Calculate the magnitude of the displacement current density \(J_d\) between the plates and the magnetic field \(B\) at a distance \(r = 0.05\ \mathrm{m}\) from the centre, assuming \(r\) is less than the plate radius.
- The displacement current \(I_d\) is equal to the conduction current \(I\) in the wires, so \(I_d = 5\ \mathrm{A}\).
- The displacement current density is \(J_d = I_d / A = 5 / 0.01 = 500\ \mathrm{A/m^2}\).
- Using the Ampère-Maxwell law for a circular path of radius \(r\): \(\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}}\). Here, \(I_{\text{enc}}\) is the displacement current enclosed by the path. Since the field is uniform between the plates, \(I_{\text{enc}} = J_d \cdot \pi r^2\).
- The left-hand side is \(B \cdot 2\pi r\). Equating both sides: \(B \cdot 2\pi r = \mu_0 J_d \pi r^2\).
- Solving for \(B\): \(B = \frac{\mu_0 J_d r}{2}\).
- Substitute values: \(\mu_0 = 4\pi \times 10^{-7}\ \mathrm{T \cdot m/A}\), \(J_d = 500\ \mathrm{A/m^2}\), \(r = 0.05\ \mathrm{m}\).
- \(B = \frac{(4\pi \times 10^{-7}) \cdot 500 \cdot 0.05}{2} = \frac{100\pi \times 10^{-7}}{2} = 50\pi \times 10^{-7}\ \mathrm{T}\).
- \(B \approx 1.57 \times 10^{-5}\ \mathrm{T}\).
Units check. \(\mu_0\) is \(\mathrm{T \cdot m/A}\), \(J_d\) is \(\mathrm{A/m^2}\), \(r\) is \(\mathrm{m}\). Product: \((\mathrm{T \cdot m/A}) \cdot (\mathrm{A/m^2}) \cdot \mathrm{m} = \mathrm{T}\). Correct.
Three quick tests
- Why does the original Ampère’s law fail for time-varying charge densities?
Answer
It implies \(\nabla \cdot \mathbf{J} = 0\), which contradicts the continuity equation when \(\partial \rho / \partial t \neq 0\). - What is the physical origin of the displacement current?
Answer
It arises from the time-varying electric field, not from the motion of charges. - How does the displacement current restore symmetry between electric and magnetic fields?
Answer
It allows a changing electric field to generate a magnetic field, mirroring Faraday’s law where a changing magnetic field generates an electric field.
Problem set for PU-204 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes checked line by line by the editor; every number in the worked example was recomputed.